In modern middle school and high school mathematics algebra is a gatekeeper course. Success in Algebra 1 in grade 8 is considered a prerequisite for college preparedness. My own first experience in Algebra 1 was at South Side Junior High School. At the time, a scientific calculator was considered high tech. Last year I taught Grade 8 algebra at The Park School in Brookline, MA. Students used graphing calculators, laptops, iPhone Apps, Java Applets, etc. Despite the incredible change in available technology, many people presume that the key skills required in an Algebra 1 course remain the same as they were when I took the course.
In recent years, as I have learned more about the use of mathematics in the workplace, I have grown increasingly disenchanted with the traditional Algebra 1 course that I received in 1979 (and I still see in existence in many schools). Topics like factoring, which I was quite good at in Grade 8, seem unconnected to any real vocation. Tools like matrices, which were left out of the course, seem to be ubiquitous in many modern, high paying STEM fields. I can think of two defensible reasons for this fact: (a) matrices can be time consuming to do by hand and (b) even with a graphing calculator they are difficult to visualize in more than the two variables.
Recently I was exploring some problems involving systems of three equations in three variables. I used Mathcad to solve the problems using matrices and then to visualize the solution on a 3-D plot. A portrait of my experiences follows. I propose that in 2010, matrices are a much more accessible topic for school algebra.
The first problem that I explored had a unique solution. Below I show how I defined matrices F and G to represent the system of three equations. Of course, I also checked the determinant of F to see if the system would yield a solution. Since |F| = 70, I solved the system by multiplication of F-1*G. In pursuing this strategy I asked Mathcad to solve the system using both numeric and symbolic evaluation. Note how the results are equivalent, but the symbolic result is a fraction instead of a decimal. Next, I used the lsolve function to see if Mathcad would replicate the same results and wrote an explanation of my findings. (Ref Fig. 1)
Some may consider the above work to be a black box solution. Mathcad has completed the calculations without showing any work. Yet, I have determined the results efficiently and explained them in writing using Mathcad’s math and text capabilities. In addition, below I use Mathcad’s graphing capabilities to further illustrate the problem. It is easy to create a 3D quick plot in Mathcad. The graph below clearly illustrates the intersection of the three planes at a single point. (Ref Fig. 2)
For contrast, I also explored a system that did not have a solution. In this case, Mathcad’s determinant function helped me to identify that the system was unsolvable. However, the more enlightening exercise was graphing the system in order to observe how the planes intersect in three-space.
Here’s the system, calculations, and explanation of my Mathcad results: (Ref Fig. 3)
Here’s the graph:
When Mathcad evaluated the determinant at zero, I knew that there would not be a solution. The symbolic results confirmed this, identifying the solution as undefined. The lsolve command, however, produced an error (shown in red text beneath my explanation). Thus, working with Mathcad to solve systems of equations offers a number of opportunities for mathematical explanation. First, the user needs to interpret the results intelligently. Do we need to debug the lsolve error or interpret its meaning? Second, there is a rich opportunity for thinking about the system by interpreting the graph. How is this graph different from the previous graph that depicted a solution? Comparing and describing the pair of graphs above can help students ground their understanding of the nature of the solution to a system of three equations in three variables.
Matrices are challenging, but they are really important in applied mathematics – they are a critical STEM topic. Engineers and scientists use matrices to solve challenging problems in many, many dimensions. Mathcad’s matrix and graphing tools offer capabilities that can help students’ explore matrices early in their school experience so that they are both prepared to use and aware of the importance of matrices. With currently available technologies matrices can be used, explored, and visualized effectively in Algebra 1 class. Systems of equations in three variables need not be avoided any longer. Matrices can be an efficient and powerful way to solve systems, with increased clarity now that we have tools to graph 3D plots. (Ref Fig. 4)